# Many-body Localization Transition: Schmidt Gap, Entanglement Length & Scaling

## TL;DR

This paper investigates the many-body localization transition, introducing the Schmidt gap as a new indicator that aligns with theoretical bounds and reveals a diverging entanglement length, contrasting with previous numerical findings.

## Contribution

The study introduces the Schmidt gap as a novel scaling measure for the transition, showing it aligns with analytical bounds and offers insights into the diverging length scale.

## Key findings

- Schmidt gap scales with a critical exponent > 2 near the transition.
- Entanglement length derived from logarithmic negativity diverges at the transition.
- Schmidt gap is less sensitive to finite size fluctuations than other quantities.

## Abstract

Many-body localization has become an important phenomenon for illuminating a potential rift between non-equilibrium quantum systems and statistical mechanics. However, the nature of the transition between ergodic and localized phases in models displaying many-body localization is not yet well understood. Assuming that this is a continuous transition, analytic results show that the length scale should diverge with a critical exponent $\nu \ge 2$ in one dimensional systems. Interestingly, this is in stark contrast with all exact numerical studies which find $\nu \sim 1$. We introduce the Schmidt gap, new in this context, which scales near the transition with a exponent $\nu > 2$ compatible with the analytical bound. We attribute this to an insensitivity to certain finite size fluctuations, which remain significant in other quantities at the sizes accessible to exact numerical methods. Additionally, we find that a physical manifestation of the diverging length scale is apparent in the entanglement length computed using the logarithmic negativity between disjoint blocks.